If I was a mathematician or a computer scientist, or even very
logically minded, I imagine it would be pretty easy to come up with a
proof that, given some small set of minimal restrictions, there would
still be no possibility of conflicts.

I don't think there is any need for such efforts. If conflicts happen
then someone needs to change a name, it's as simple as that.

As long as the system is reasonable (e.g. basing it on domain names)
then in practice the change of conflicts is small, and simplicity wins out.